arXiv:1302.0374v4 [math.AG] 2 Sep 2014
AND 4-FOLDS OF GENERAL TYPE, III
JUNGKAI A. CHEN AND MENG CHEN
Abstract. Nonsingular projective 3-folds V of general type can be naturally classified into 18 families according to the pluricanoni-cal section index δ(V ) := min{m|Pm≥ 2} since 1 ≤ δ(V ) ≤ 18 due to our previous series (I, II). Based on our further classification to 3-folds with δ(V ) ≥ 13 and an intensive geometrical investigation to those with δ(V ) ≤ 12, we prove that Vol(V ) ≥ 1
1680 and that the pluricanonical map Φmis birational for all m ≥ 61, which greatly improves known results. An optimal birationality of Φm for the case δ(V ) = 2 is obtained. As an effective application, we study projective 4-folds of general type with pg≥ 2 in the last section.
1. Introduction
One of the fundamental aspects of birational geometry is to under-stand the behavior of the natural pluricanonical map Φm of any variety
for any m ∈ Z>0. The induced fibrations possibly reduce the studies
to lower dimensional situations. Varieties of general type, which are those with birational pluricanonical maps Φm for sufficiently large m,
are therefore considered as the basic building blocks of varieties. For varieties of general type, a key problem is to find an effective integer m > 0 so that Φm is birational. The remarkable theorem of
Hacon and McKernan [17], Takayama [31], and Tsuji [32] says that there is a constant c(n) so that Φm is birational for all n-dimensional
varieties of general type and for all m ≥ c(n). However, these constants are explicitly known only when n ≤ 3.
In fact, the problem is almost equivalent to find a practical lower bound of the canonical volume which computes the rate of growth of plurigenera, or equivalent to find m0 such that plurigenus Pm0 is
sufficiently large. One may also refer to the nice survey article Hacon– McKernan [18] for various boundedness results in birational geometry. The motivation of this series is to study birational geometry of 3-folds and higher dimensional varieties of general type. The main purpose is to investigate the following:
The first author was partially supported by NCTS/TPE and National Science Council of Taiwan. The second author was supported by National Natural Science Foundation of China (#11171068, #11121101, #11231003) and Doctoral Fund of Ministry of Education of China (#20110071110003).
Open problem 1.1. Find optimal constants v3 ∈ Q>0 and b3 ∈ Z>0
so that, for all nonsingular projective 3-folds V of general type, i. Vol(V ) ≥ v3 and
ii. Φm is birational for all m ≥ b3.
Recall that we have proved the following:
Theorem 1.2. ([7, Theorems 1.1, 1.2]) Let V be a nonsingular projec-tive 3-fold of general type. Then
1. Vol(V ) ≥ 1
2660;
2. there exists a positive integer m0(V ) ≤ 18 so that Pm0 ≥ 2;
3. the pluricanonical map Φm is birational onto its image for all
m ≥ 73.
For more results on explicit birational geometry of 3-folds of general type, one may refer to our previous papers [6, 7].
In order to formulate our main statements of this article, we need to recall some general results and introduce some definition. Given a projective variety V of general type, there exists a minimal model X birational to V (cf. [2]). Thanks to the Riemann-Roch formula and Vanishing Theorem, Vol(V ) = Kdim X
X . Notice that in dimension three
or higher, a minimal model may have singularities. Hence Kdim X
X is
just a positive rational number.
A minimal model has at worst terminal singularities. In dimension three, terminal singularities was classified by Mori. A three dimen-sional terminal singularity is one of the following: a terminal quotient singularity of type 1
r(1, −1, b) for some b relatively prime to r which we
usually denote it as (b, r) for short, an isolated cDV point, a quotient of an isolated cDV point. It is well-known to experts that a three di-mensional terminal point can be deformed into a collection of terminal quotient singularities, which is called basket of singularities. An impor-tant feature of three dimensional birational geometry is the Singular Riemann-Roch formula due to Reid [29]
χ(X, mKX) =
m(m − 1)(2m − 1)K3
X
12 + (1 − 2m)χ(X, OX) + lm,
where lm denotes the contribution of singularities which can be
com-puted by baskets. It follows that all plurigenera and hence canonical
volume of a minimal 3-fold X are completely determined by P2(X),
χ(X, OX) and baskets of singularities BX, of which we called such a
triple the weighted basket of X. For the basic properties of weighted baskets, one may refer to [6, Section 3]. Since our problems are bira-tional in nature, the studies of nonsingular threefold V is equivalent to the studies of its minimal model X. In particular, we may and do
consider the weighted basket of V as the weighted basket of its minimal
model X. 1
Next, we would like to define the pluricanonical section index (or, in short, the ps-index)
δ(V ) := min{m|m ∈ Z>0, Pm(V ) ≥ 2},
which is clearly a birational invariant. By Theorem 1.2, we have δ(V ) ≤ 18 for any 3-fold V of general type. Note that 3-folds V with δ(V ) = 1 (i.e., pg(V ) ≥ 2) have been intensively studied in [10, 11] where optimal
results are realized. Threefolds of general type with δ(V ) ≥ 2 are far from being clear. Sometimes we use the symbol δ(X) directly since X is birationally equivalent to V .
Example 1.3. The “worst” known minimal 3-fold is the weighted hyper-surface X := X46 ⊂ P(4, 5, 6, 7, 23) (cf. [15]) which has the
invariants: δ(X) = 10 and Vol(X) = K3
X = 4201 . Also Φ26 is not
birational.
In this paper, we mainly investigate projective 3-folds of general type with δ(V ) ≥ 2. Our main results are as follows.
Theorem 1.4. (=Theorem 5.1) Let V be a nonsingular projective 3-fold of general type with δ(V ) ≥ 13. Then its weighted basket B = {BV, P2(V ), χ(OV)} belongs to one of the types in Tables F–0, F–1,
F–2 in Appendix and the following is true:
(1) δ(V ) = 18 if and only if B(V ) = {B2a, 0, 2};
(2) δ(V ) 6= 16, 17;
(3) δ(V ) = 15 if and only if B(V ) belongs to one of the types in Table F–1;
(4) δ(V ) = 14 if and only if B(V ) belongs to one of the types in Table F–2;
(5) δ(V ) = 13 if and only if B(V ) = {B41, 0, 2},
where B2a and B41 can be found in Table F–0
Some other results for 3-folds with large δ(V ) are given in Section 4. For example, one has
Corollary 1.5. (=Corollary 4.8) Let V be a nonsingular projective 3-fold of general type with Vol(V ) < 3361 . Then δ(V ) ≥ 8.
We also prove the following:
Theorem 1.6. Let V be a nonsingular projective 3-fold of general type. Then
(1) Φm is birational for all m ≥ 61;
1Even though minimal models are not necessarily unique, it is known that two
birational minimal models are connected by flops (cf. [21]). Together with the fact that a 3-dimensional flop preserves singularity types (cf. [23]), it follows that baskets of V is independent of choices of minimal models.
(2) Vol(V ) ≥ 1
1680. Furthermore, Vol(V ) =
1
1680 if and only if
B(V ) = {B7a, 0, 2} or {B36a, 0, 2}, where B7a and B36a can be
found in Table F–2
A direct by-product of our method is the following:
Corollary 1.7. Let V be a nonsingular projective 3-fold of general type with pg(V ) = 1. Then
(1) Vol(V ) ≥ 1 75;
(2) Φm is birational for all m ≥ 18.
In the second part of this paper we prove some optimal results on 3-folds with δ(V ) = 2.
Theorem 1.8. Let V be a nonsingular projective 3-fold of general type with δ(V ) ≤ 2. Then
(1) Φm is birational for all m ≥ 11;
(2) If Φ10 is not birational, then 0 ≤ χ(OV) ≤ 3 and |2KV| is
composed of a rational pencil of (1, 2) surfaces. Furthermore, #{B(V )} < +∞ and the initial basket B0 of B
V belongs to one
of the types in Tables II–1, II–2, II–3 in the Appendix.
The following examples show that our results in Theorem 1.8 are optimal.
Example 1.9. (Iano-Fletcher [15, P. 151, P. 153])
(1) General weighted complete intersections X22 ⊂ P(1, 2, 3, 4, 11)
and X6,18 ⊂ P(2, 2, 3, 3, 4, 9) both have ps-index δ = 2. Since
both X22 and X6,18have non-birational 10-canonical map,
The-orem 1.8(1) is optimal.
(2) The 3-fold X22 corresponds to No. 1 in Table II–1 with χ = 0
and X6,18 belongs to No. 11 (with t = 1) in Table II–1.
Remark 1.10. Theorem 1.8 is parallel to main results in [10]. We have similar statements to Theorem 1.8 for 3-folds with δ(V ) ≥ 3. We omit them since we are not sure whether they are optimal or not.
In the last part we study projective 4-folds. The main result is the following:
Theorem 1.11. (=Theorem 8.2) Let V be a nonsingular projective 4-fold of general type. Then,
(i) when pg(V ) ≥ 2, Φ|mKV| is birational for all m ≥ 35;
(ii) when pg(V ) ≥ 19, Φ|mKV| is birational for all m ≥ 18.
This paper is organized as follows. In Section 2, we start with general setting on rational maps on varieties of general type and review some known useful inequalities. Then we list several basic lemmas on 3-folds. In Section 3, we improve our technique used in [7] to bound K3
from below. Applying our basket analysis developed in [6], we obtain an effective function v(x) in Section 4 so that K3
X ≥ v(δ(X)) for any
given minimal 3-fold X. Section 5 is devoted to compiling the clean list for B(X) with δ(X) ≥ 13. Then, in Section 6, we are able to study the birationality of Φm. Section 7 is dedicated to classifying 3-folds
with δ = 2. Finally we study nonsingular projective 4-folds of general type with pg ≥ 2 in Section 8. All subsidiary tables are presented in
the Appendix.
Throughout we work over any algebraically closed field k of charac-teristic 0. We are in favor of the following symbols:
◦ “∼” denotes linear equivalence or Q-linear equivalence; ◦ “≡” denotes numerical equivalence;
◦ “|A| |B|” means that |B| ⊇ |A| + fixed effective divisors. 2. Preliminaries
We begin with the general setting on rational maps defined by some sub-linear system of the pluricanonical system |mK| on varieties of general type. Let V be any nonsingular projective variety of general type with dimension n ≥ 3. According to the Minimal Model Program, V has a minimal model (see e.g. [22], [24], [2] and [30]). From the point of view of birational geometry, we may always consider the rational map on minimal varieties of general type. A minimal model X is a normal projective variety with a nef canonical divisor KX and with Q-factorial
terminal singularities.
2.1. The rational map ΦΛ for Λ ⊂ |m0K|. Let X be a minimal
projective variety of general type on which Pm0(X) ≥ 2 for a positive
integer m0. Let Λ ⊂ |m0KX| be a positive dimensional linear system.
Fix an effective Weil divisor Km0 ∼ m0KX on X. Take successive
blow-ups π : X′ → X along nonsingular centers, such that the following
conditions are satisfied: (i) X′ is smooth;
(ii) the moving part of π∗(Λ) is base point free and so that g :=
ΦΛ◦ π is a non-constant morphism;
(iii) π∗(K
m0)∪{π −exceptional divisors} has simple normal crossing
supports.
Sometimes we will take further blow-ups so that π satisfies some more conditions, which will be specified explicitly.
We have a morphism g : X′ −→ Φ
Λ(X) ⊆ PN. Let X′ f
−→ Γ −→s
ΦΛ(X) be the Stein factorization of g. We have the following
X′ f // π g '' ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Γ s X ΦΛ // ΦΛ(X)
We may write m0KX′ =Q π∗(m0KX) + Eπ,m0 where Eπ,m0 is an
effec-tive π-exceptional Q-divisor. Denote by Mm0 (resp. MΛ) the
mov-able part of |m0KX′| (resp. π∗Λ). Set dm0 := dim Φm0(X) (resp.
dΛ := dim Γ). The Bertini Theorem implies that the general
mem-ber of the moving part MΛ of π∗(Λ) is irreducible whenever dΛ ≥ 2
and, otherwise, MΛ ≡ aΛF , where aΛ := deg f∗OX′(MΛ) and F is a
general fiber of f . We set θΛ:=
(
1, if dΛ≥ 2;
aΛ, if dΛ= 1.
Recall our definition in [7, Definition 2.4], the generic irreducible ele-ment Σ of π∗(Λ) is defined as follows:
ΣΛ :=
(
the general member of the moving part of π∗(Λ), if d Λ ≥ 2;
F, if dΛ = 1.
By the above setting, we always have
m0π∗(KX) ∼Q θΛΣΛ+ EΛ′
for some effective Q-divisor E′
Λ on X′.
Convention. Whenever we are working on the complete linear system |m0KX|, we will use parallel notations such as dm0, θm0, · · · (or even
just d, θ, · · · , for simplicity).
We discuss the special case with dΛ = 1. Clearly the general fiber
F is nonsingular projective of dimension dim(X) − 1. Replace X′ by
its birational model, we may assume that there is a birational
contrac-tion morphism σ : F −→ F0 onto a minimal model F0. We have the
following “canonical restriction inequality”:
Lemma 2.1. Keep the above settings. Suppose that dΛ = 1. The
following holds: (i) if b := g(Γ) > 0, then π∗(K X)|F ∼ σ∗(KF0); (ii) if b = 0, then π∗(KX)|F ≥ θΛ m0+ θΛ σ∗(KF0).
Assume Γ ∼= P1. Choose a sufficiently large and divisible integer m so that both |mπ∗(K
X)| and |mKF0| are base point free. By Kawamata’s
extension theorem [20, Theorem A], we have the surjective map: H0(X′, mθΛ(KX′ + F )) −→ H0(F, mθΛKF).
Since |m(θΛ+m0)KX′||mθΛ(KX′+F )|, Mov|mθΛKF| = |mθΛσ∗(KF0)|
and |m(θΛ + m0)π∗(KX)| = |Mm(θΛ+m0)|, we obtain the following
in-equality:
m(θΛ+ m0)π∗(KX)|F = Mm(θΛ+m0)|F ≥ mθΛσ
∗(K F0),
which implies (ii).
2.2. Key inequalities on 3-folds. Let X be minimal 3-fold of gen-eral type. Assume that Λ ⊂ |m0KX| is a linear system of positive
dimension. As in 2.1, we obtain an induced fibration f : X′ −→ Γ.
Pick a generic irreducible element S of |m0KX′|. Let |G| be a given
base point free linear system on S. Pick a generic irreducible element
C of |G|. Since π∗(K
X)|S is nef and big, Kodaira’s lemma implies
that π∗(K
X)|S ≥ βC for some rational number β > 0. Then, by [7,
Inequality (2.1)], one has
KX3 ≥ θβ m0
ξ (2.1)
where ξ := (π∗(K
X) · C)X′. Besides, by [7, Remark 2.12], one has
ξ ≥ deg(KC) 1 + m0 θ + 1 β . (2.2)
For any positive integer m so that αm := (m − 1 − mθ0 − β1)ξ > 1, by
Chen–Zuo [13, Theorem 3.1], one has
ξ ≥ deg(KC) + ⌈αm⌉
m . (2.3)
We have a stronger form of Inequality (2.3) when C is “even”: Lemma 2.2. Under the above situation, if C is an even divisor on S (i.e. 12C ∈ Pic(S)), then, for any m > 0 so that αm > 0, one has
ξ ≥ deg(KC) + 2⌈
1 2αm⌉
m . (2.4)
Proof. We refer to the proof for Chen–Zuo [13, Theorem 3.1]. The key point is to estimate deg(D) where D = ⌈Q⌉|C and Q is a Q-divisor on
S with (Q · C) = αm. Since deg(D) ≥ αm > 0 and deg(D) is even, we
naturally have
deg(D) = 2(⌈Q⌉ · 1
2C) ≥ 2⌈ 1 2αm⌉
where we note that (⌈Q⌉ · 12C) is a positive integer. Clearly the rest of the proof of Chen–Zuo [13, Theorem 3.1] implies Inequality (2.4).
When dΛ= 1, Lemma 2.1(ii) implies the following:
ξ = (π∗(KX) · C)X′ ≥
θ m0+ θ
(σ∗(KF0) · C)F. (2.5)
2.3. Other useful Lemmas.
Lemma 2.3. (see Ma¸sek [25, Proposition 4] or [12, Lemma 2.6]) Let S be a nonsingular projective surface. Let L be a nef and big Q-divisor on S satisfying the following conditions:
(1) L2 > 8.
(2) (L · Cx) ≥ 4 for all irreducible curves Cx passing through any
very general point x ∈ S.
Then the linear system |KS+ ⌈L⌉| separates two distinct points in very
general positions. Consequently, |KS+ ⌈L⌉| gives a birational map.
Lemma 2.4. Let σ : S −→ S0 be a birational contraction from a
nonsingular projective surface S of general type onto the minimal model S0. Assume that (KS20, pg(S0)) 6= (1, 2) and that C is a moving curve
on S. Then (σ∗(KS0) · C) ≥ 2.
Proof. When K2
S0 ≥ 2, this is due to Hodge index theorem. When
(K2
S0, pg(S0)) = (1, 0), this is due to Miyaoka [26, Lemma 5]. When
(K2
S0, pg(S0)) = (1, 1), (σ
∗(K
S0) · C) = 1 implies KS0 ≡ σ∗C by Hodge
index theorem. According to Bombieri [3], we know that S0 is simply
connected. Thus KS0 ∼ σ∗C, which is impossible since |KS0| is not
movable.
Lemma 2.5. Let σ : S −→ S0 be the birational contraction onto the
minimal model S0 from a nonsingular projective surface S of general
type. Assume that (K2
S0, pg(S0)) 6= (1, 2) and that ˜C is a curve on S
passing through very general points. Then (σ∗(K
S0) · ˜C) ≥ 2.
Proof. In fact, by the projection formula, this is equivalent to see (KS0·
C0) ≥ 2 for any curve C0 ⊂ S0 passing through very general points of
S0.
On the contrary, let us assume (KS0·C0) ≤ 1. Then g(C0) ≥ 2 implies
C2
0 ≥ 1. The Hodge index theorem says KS20 = 1 and KS0 ≡ C0. Recall
that S0 is not a (1,2) surface. So S0 must be either a (1, 0) surface or
a (1, 1) surface. If (K2
S0, pg(S0)) = (1, 0), then q(S0) = 0 and the torsion element θ :=
KS0− C0 is of order ≤ 5 (see Reid [27]) and h
0(S
0, C0) = 1. Thus there
are at most finite number of such curves on S0 since #Tor(S0) ≤ 5,
which is absurd by the choice of C0.
If (K2
S0, pg(S0)) = (1, 1), then q(S0) = 0 and KS0 ∼ C0since Tor(S0) =
0 by Bombieri [3, Theorem 15] and thus C0is the unique canonical curve
2.4. The birationality principle.
Definition 2.6. Pick two different generic irreducible elements S′, S′′
(resp. C′, C′′) in |M
m0| (resp. in |G|).
(1) We say that |mKX′| distinguishes S′ and S′′ if Φ|mK X′|(S
′) 6=
Φ|mKX′|(S
′′).
(2) We say that |mKX′| distinguishes C′ and C′′ if Φ|mK X′|(C
′) 6=
Φ|mKX′|(C
′′).
We will apply the useful, but technical theorem in Chen-Zuo [13] for the birationality of Φm.
Theorem 2.7. (see Chen-Zuo [13, Theorem 3.1] or [7, Theorem 2.11, Part 2]) Keep the same notations as above. Assume that, for some m > 0, |mKX′| distinguishes S′ and S′′, C′ and C′′for generic S′ 6= S′′,
C′ 6= C′′. Then Φ
m is birational under one of the following conditions:
(i) αm > 2;
(ii) αm > 1 and C is not hyper-elliptic.
3. The lower bound of K3 in terms of m
0
In the study of 3-dimensional explicit birational geometry, a chal-lenging problem is to determine whether a given weighted basket B is geometric, i.e. equal to BX for some 3-fold X or not. By exploiting
geometric properties, one might be able to have a better estimation of the lower bound of K3
X, and hence exclude some non-geometric formal
baskets. In fact, in [7, 2.19∼2.31], we already proved some effective inequalities for K3
X. We shall go further along this direction in this
section
Let X be a minimal 3-fold of general type. Assume Pm0(X) ≥ 2.
Mostly we will take Λ = |m0KX|. Keep the settings in 2.1 and 2.2.
3.1. The case dm0 = 3.
If we take |G| to be |S|S|, then β = m10. It is known, from [7, 2.19], that
deg(KC) ≥ 6, ξ ≥ 3m100+2 and KX3 ≥ mξ2
0. Take m = 5m0 + 4, · · · , (2t +
1)m0 + 2t, successively. Then, by (2.3), one has ξ ≥ 5m170+4,7m240+6,
· · · , 7t+3
(2t+1)m0+2t respectively. Taking the limit, we obtain ξ ≥
7 2m0+2. Therefore KX3 ≥ 7 2m2 0(m0+ 1) . (3.1)
In fact, for each small m0, the explicit lower bound of K3 can be
slightly improved by the same trick and here is the result: Table A1
m0= 2 3 4 5 6 7 8 ξ ≥ 4/3 1 3/4 5/8 1/2 6/13 2/5 K3 ≥ 1/3 1/9 3/64 1/40 1/72 6/637 1/160 m0= 9 10 11 12 13 14 15 ξ ≥ 4/11 1/3 3/10 5/18 1/4 6/25 2/9 K3 ≥ 4/891 1/300 3/1210 5/2592 1/696 3/2450 2/2025 3.2. The case dm0 = 2. If we take |G| = |S|S|, then β ≥ Pm0−2
m0 . By Inequality (2.3), one has
ξ ≥ 2m20+1. Take m = 3m0+2, 5m0+4, · · · , (2t+1)m0+2t successively.
One gets from Inequality (2.3) that ξ ≥ 3m40+2,5m70+4, · · · ,(2t+1)m3t+10+2t.
Taking the limit, we have ξ ≥ 3
2m0+2. By Inequality (2.1), we have KX3 ≥ 3(Pm0 − 2) 2m2 0(m0+ 1) ≥ 3 2m2 0(m0+ 1) . (3.2)
In fact, we have the following estimation for each small m0, which
slightly improves [7, Table A]:
Table A2 m0= 2 3 4 5 6 7 8 ξ≥ 1/2 2/5 1/3 1/4 2/9 1/5 1/6 K3≥ 1/8 2/45 1/48 1/100 1/162 1/245 1/384 m0= 9 10 11 12 13 14 15 ξ≥ 2/13 1/7 1/8 2/17 1/9 1/10 2/21 K3≥ 2/1053 1/700 1/968 1/1224 1/1521 1/1960 2/4725 Under the same situation, if there exists a number m1 > 0 such that
dm1 = 3, then, since (m1π ∗(K X)|F · C) ≥ 2, we have ξ ≥ m21. Thus Inequality (2.1) reads: KX3 ≥ 2(Pm0 − 2) m2 0m1 ≥ 2 m2 0m1 . (3.3) 3.3. The case dm0 = 1, b = g(Γ) > 0.
We have S = F by definition. Pick a very large number l > 0. Take |G| := |lσ∗(K
F0)| which is base point free by the surface theory. By
definition, we have θ ≥ Pm0 ≥ 2. Since π
∗(K
X)|F ∼ σ∗(KF0) by Lemma
2.1(i), we see β = 1
l and thus Inequality (2.1) implies
KX3 ≥ Pm0 m0 ·1 l · lK 2 F0 ≥ Pm0 m0 . (3.4) 3.4. The case dm0 = 1, b = 0.
By Lemma 2.1(ii), we have KX3 ≥ θ m0 π∗(KX)|2F ≥ θ3 m0(m0+ θ)2 · KF20. (3.5)
We will choose suitable linear system |G| on F depending on the numerical type of F . From the surface theory, we know that either K2 F0 ≥ 2 or (K 2 F0, pg(F )) = (1, 2), (1, 1), (1, 0). Subcase 3.4.1. K2 F0 ≥ 2. Inequality (3.5) implies KX3 ≥ 2θ 3 m0(m0+ θ)2 . (3.6) Subase 3.4.2. (K2 F0, pg(F0)) = (1, 2).
Take |G| := Mov|KF|. Then C, as a generic irreducible element of
|G|, is a smooth curve of genus 2 (see [1]). By Lemma 2.1(ii), we have β = m0θ+θ ≥ 1 m0+1. Inequality (2.2) implies ξ ≥ θ m0+θ. Take m = ⌊ 3m0+3θ θ ⌋ + 1 > 3m0+3θ θ .
Then, since αm ≥ (m − 1 − mθ0 − β1)ξ > 1, Inequality (2.3) gives ξ ≥ 4 ⌊3m0+3θθ ⌋+1 ≥ 4θ 3m0+4θ. Inductively, take m = ⌊ (1+23(4t−1))m 0+3·4t−1θ 4t−1θ ⌋ + 1, one gets ξ ≥ (1+2 4tθ 3(4t−1))m0+4tθ and hence ξ ≥ 3θ
2m0+3θ by taking the limit.
Thus we have KX3 ≥ 3θ 3 m0(m0+ θ)(2m0+ 3θ) ≥ 3 m0(m0+ 1)(2m0+ 3) . (3.7)
A similar calculation leads to the following better estimation for smaller m0: Table A3 m0= 2 3 4 5 6 7 8 ξ≥ 1/2 1/3 2/7 1/4 1/5 2/11 1/6 K3≥ 1/12 1/36 1/70 1/120 1/210 1/308 1/432 m0= 9 10 11 12 13 14 15 ξ≥ 1/7 2/15 1/8 1/9 2/19 1/10 1/11 K3≥ 1/630 1/825 1/1056 1/1404 1/1729 1/2100 1/2640 Subcase 3.4.3. (K2 F0, pg(F0)) = (1, 1). Since |σ∗(K
F0)| is not moving, we have to take |G| := |2σ
∗(K
F0)| which
is base point free by the surface theory. Naturally the generic irre-ducible element C of |G| is even and deg(KC) = 6.
By Lemma 2.1(ii), we have β = 2m0θ+2θ. Take m = ⌊3m0+3θ
θ ⌋ + 1.
Since ξ > 0, we have αm > 0. Thus Lemma 2.2 implies ξ ≥ 3m8θ0+4θ.
Thus Inequality (2.1) reads
KX3 ≥ 4θ
3
m0(m0+ θ)(3m0+ 4θ)
. (3.8)
For each small m0, we have the following better estimation:
m0 = 2 3 4 5 6 7 8 ξ≥ 6/7 2/3 1/2 4/9 3/8 1/3 2/7 K3≥ 1/14 1/36 1/80 1/135 1/224 1/336 1/504 m0 = 9 10 11 12 13 14 15 ξ≥ 4/15 6/25 2/9 1/5 4/21 14/79 1/6 K3≥ 1/675 3/2750 1/1188 1/1560 1/1911 1/2370 1/2880 Subcase 3.4.4. (K2 F0, pg(F0)) = (1, 0).
Modulo further birational modification, we may assume that Mov|2KF|
is base point free. Take |G| = Mov|2KF|. By Catanese-Pignatelli [4],
the generic irreducible element C of |G| is a smooth curve of genus ≥ 3.
By Lemma 2.1(ii), we have β = θ
2m0+2θ ≥ 1 2m0+2. Lemma 2.4 implies ξ ≥ mθ 0+θ · (σ ∗(K F0) · C) ≥ 2θ m0+θ. Thus we have KX3 ≥ θ 3 m0(m0+ θ)2 . (3.9)
Of course, for each small m0, one might get slightly better estimation
for ξ and K3 X.
Variant 3.4.5. If there exists a positive integer m1 such that Pm1 ≥ 2
and that |m0KX′| and |m1KX′| are not composed with the same pencil.
We may take |G| = |Mm1|F| and then we have β =
1
m1. Thus Inequality
(2.1) and Lemma 2.4 imply
KX3 ≥ 2θ 2 m0 m0m1(m0+ θm0) , (3.10) provided that (K2 F0, pg(F0)) 6= (1, 2).
3.5. Some other inequalities.
Corollary 3.1. Let X be a minimal 3-fold of general type. Assume Pm0 = 2. Keep the same notation as above. Suppose that the general
fiber F of the induced fibration from Φm0 is not a (1, 2) surface, and
that Pm1 ≥ 2 for some integer m1 > 0. Then
KX3 ≥ min{ (Pm1 − 1) 3 m1(m1 + Pm1 − 1)2 , 2 m0m1(m0+ 1) }.
Proof. If |m0KX′|, |m1KX′| are composed with the same pencil, then
both |m0KX′| and |m1KX′| induce the same fibration f : X′ −→ Γ.
Consider ˜Λ = |m1KX′|. Then, θm1 ≥ Pm1 − 1. Since F is not a (1,2)
surface and by comparing Inequality 3.4, 3.6, 3.8 and 3.9, we have
KX3 ≥ (Pm1 − 1)
3
m1(m1+ Pm1 − 1)2
.
Suppose that |m0KX′|, |m1KX′| are not composed with the same
pencil. We have β = 1
m1. Then we have Inequality (3.10) as in Variant
Now we are able to study the more restricted case:
Proposition 3.2. Let X be a minimal 3-fold of general type. Assume that Pm0(X) ≥ 4 and dm0 = 2, then
KX3 ≥ min { 8 m0(m0+ 2)2 , 6 m2 0(m0 + 2) }.
Proof. We need to study the image surface W′ of X′ through the
mor-phism Φ|m0KX′|. In fact, we have the Stein factorization
Φm0 := Φ|m0KX′| : X
′ −→ Γf −→ Ws ′ ⊂ PPm0−1.
Denote by H′ a very ample divisor on W′ such that M
m0 ∼ Φ
∗ m0(H
′).
Furthermore one has Mm0|S ≡ ˜am0C for a general member S ∈ |Mm0|
and the integer ˜am0 ≥ deg(s) deg(W
′) ≥ deg(W′) ≥ P
m0 − 2, where C
is a general fiber of f . Set |G| := |Mm0|S|.
Case 1. ˜am0 ≥ 3.
We have β ≥ m30. Inequality (2.2) implies ξ ≥ 4m60+3. Take m = 2m0+ 2. Then Inequality (2.3) gives ξ ≥ m02+1. Take m = ⌊11m60+9⌋+1.
Since αm > (11m60+9 − 1 − m0 − 1β)ξ ≥ 1, Inequality (2.3) implies
ξ ≥ 11m240+15. Thus, we have KX3 ≥ 72 m2 0(11m0+ 15) . (3.11) Case 2. ˜am0 = 2.
Automatically we have Pm0 = 4, which also implies that deg(W
′) = 2
and deg(s) = 1. Recall that an irreducible surface (in P3) of degree 2
is one of the following surfaces (see, for instance, Reid [28, p. 30, Ex. 19]):
(a) W′ is the cone F
2 obtained by blowing down the unique section
with the self-intersection (−2) on the Hirzebruch ruled surface F2;
(b) W′ ∼= P1× P1.
Case 2.a. W′ = F
2.
Replacing by its birational model, we may assume that Φm0 factors
through the minimal resolution F2 of W′. So we have the factorization
of Φm0 : X
′ −→ Fh 2
ν
−→ W′ where h is a fibration and ν is the minimal
resolution of W′. Set ˆH = ν∗(H′). We know that H′2 = 2 and hence
ˆ
H2 = 2. Noting that ˆH is nef and big on F
2, we can write
ˆ
H ∼ µG0+ nT
where µ and n are integers, G0 denotes the unique section with G20 =
−2, and T is the general fiber of the ruling on F2. The property of
ˆ
pr : F2 −→ P1 be the ruling. Set ˜f := pr ◦ h : X′ −→ P1, which is a
fibration with connected fibers. Denote by F a general fiber of ˜f . We have
Mm0 ∼ Φ
∗ m0(H
′) = h∗( ˆH) ≥ 2F.
Let Λ = |2F ||m0KX′|. Clearly we have θΛ = 2, dΛ = 1 and b = 0.
By Inequalities (3.6), (3.7), (3.8) and (3.9), we have
KX3 ≥ 8
m0(m0+ 2)2
. (3.12)
Case 2.b. W′ = P1× P1.
We have an induced fibration f : X′ −→ W′ = P1 × P1. Since a
very ample divisor H′ on W′ with H′2 = 2 is linearly equivalent to
L1+L2 = q1∗(point)+q2∗(point) where q1, q2 are projections from P1×P1
to P1 respectively. Set ˜f
i := qi ◦ f : X′ −→ P1, i = 1, 2. Then ˜f1 and
˜
f2 are two fibrations onto P1. Let F1 and F2 be general fibers of ˜f1 and
˜
f2, respectively. Then F1 ∩ F2 is simply a general fiber C of f . We
will estimate ξ in an alternative way. In fact, the following argument is similar to the proof of [13, Theorem 3.1].
Since ˜am0 = 2, we have S|S ∼ 2C. On the other hand, we have
S ≥ F1+ F2. Modulo further birational modifications, we may write
m0π∗(KX) ≡ F1+ F2 + Hm′ 0 where H
′
m0 is an effective Q-divisor with
simple normal crossing supports. For any integer m > m0 + 1, we
consider the linear system
|KX′+ ⌈(m − m0− 1)π∗(KX)⌉ + F1+ F2||mKX′|.
Since (m − m0 − 1)π∗(KX) + F2 is nef and big, Kawamata-Viehweg
vanishing ([19, 33]) gives the surjective map:
H0(KX′ + ⌈(m − m0− 1)π∗(KX)⌉ + F2+ F1)
−→ H0(F1, KF1 + ⌈(m − m0− 1)π
∗(K
X)⌉|F1+ C).
Using the vanishing theorem again, one gets the surjective map: H0(F1, KF1 + ⌈(m − m0− 1)π ∗(K X)|F1⌉ + C) −→ H 0(C, K C + ˆDm) where ˆDm := ⌈(m − m0− 1)π∗(KX)|F1⌉|C with deg( ˆDm) ≥ (m − m0 − 1)ξ.
When m is large enough so that deg( ˆDm) ≥ 2, the above two surjective
maps directly implies
mξ ≥ deg(KC) + deg( ˆDm) ≥ 2 + ⌈(m − m0− 1)ξ⌉. (3.13)
In particular, we have ξ ≥ m02+1.
Take m = 2m0 + 3. Then (m − m0− 1)ξ > 2 and Inequality (3.13)
gives ξ ≥ 2m50+3.
Assume m0 > 1 and take m = 2m0 + 2. One gets ξ ≥ 2m50+2. Take
m = ⌊7m0+12 5 ⌋ = ⌊ 7m0+7 5 ⌋ + 1 > 7m0+7 5 , one has ξ ≥ 4 m ≥ 20 7m0+12.
Inductively, take m = ⌊(2+53(4t−1))m0+2+103(4t−1)
5·4t−1 ⌋ for t ≥ 1, one has
ξ ≥ (2+5 5·4t
3(4t−1))m0+2+103 (4t−1). We have ξ ≥
3
m0+2 by taking the limit and
hence KX3 ≥ 1 m0 · (π∗(KX)|S)2 ≥ 2 m2 0 · ξ ≥ 6 m2 0(m0+ 2) . (3.14)
We conclude the statement by comparing 3.11, 3.12 and 3.14.
Corollary 3.3. Let X be a minimal 3-fold of general type. The fol-lowing holds: KX3 ≥(min { 8 m0(m0+2)2, 7 2m2 0(m0+1)}, when Pm0 ≥ 4; 3 2m2 0(m0+1), when Pm0 = 3.
Proof. When Pm0 ≥ 4, dm0 = 3, 2, 1 and the inequality follows from
comparing Inequality (3.1), Proposition 3.2, Inequalities (3.4, 3.6, 3.7, 3.8, 3.9) (with θm0 = 3), respectively.
When Pm0 = 3, dm0 = 2, 1 and the inequality follows immediately
by comparing Inequality (3.2) with Inequalities (3.4, 3.6, 3.7, 3.8, 3.9)
(with θm0 = 2).
4. Threefolds with δ(V ) ≤ 12
The purpose of this section is to prove the following sharper bounds: Theorem 4.1. Let X be a minimal projective 3-fold of general type
with 2 ≤ δ(X) ≤ 12. Then K3
X ≥ v(δ(X)), where the function v(x) is
defined as follows:
x 2 3 4 5 6 7
v(x) 1/14 1/36 1/90 1/135 1/224 1/336
x 8 9 10 11 12 −−
v(x) 1/504 1/675 3/2750 1/1188 1/1560 −−
We are going to estimate the lower bound of the volume, case by case, for a given δ. The discussion here relies on those formulae in [6, (3.6)-(3.12)].
Proposition 4.2. If P2(X) ≥ 2, then KX3 ≥ 141.
Proof. Set m0 = 2. By Table A1, Table A2, Inequalities (3.4) and
(3.6), Table A3, Table A4 and Corollary 3.3, we have K3
X ≥ 141 unless
P2 = 2, d2 = 1, b = 0 and F is of type (1, 0).
In the remaining case, we have that χ(OX) = 1 by [7, Lemma 2.32].
By [7, Lemma 3.2], one has P4 ≥ 2P2 ≥ 4. If d4 ≥ 2, then KX3 ≥ 121
by Inequality (3.10) (with m0 = 2, m1 = 4, θ2 = 1). If d4 = 1, then
|2KX′| and |4KX′| are composed with the same pencil. Thus we have
K3
X ≥ 19627 > 1
Proposition 4.3. If P3(X) ≥ 2, then KX3 ≥ 361.
Proof. Take m0 = 3 and Λ = |3KX′|. One has KX3 ≥ 1
36 by Table A1,
Table A2, Inequalities (3.4), (3.6), Table A3, Table A4 and Corollary 3.3 (m0 = 3) unless we are in Subcase 3.4.4 with P3 = 2. That is,
P3 = 2, d3 = 1, b = 0 and F is of type (1, 0). Again, χ(OX) = 1. Thus,
for any m ≥ 2, [7, Lemma 3.2] implies Pm+2 ≥ Pm+ P2.
By Corollary 3.1, if P4 ≥ 3 (resp. P5 ≥ 3), then KX3 ≥ 241 (resp. 1 30).
Suppose that both P4 ≤ 2 and P5 ≤ 2, then P5 = 2 and P2 = 0. By [6,
(3.6)], n0
1,2 = 5 − 8 + P4 < 0, which is a contradiction. Hence either P4
or P5 ≥ 3 in this case and we are done.
Proposition 4.4. If P4(X) ≥ 2, then KX3 ≥ 901.
Proof. Similarly, we have K3
X ≥ 801 unless P4 = 2, b = 0 and F is
of (1, 0) type. In fact, in this situation, we have at least KX3 ≥ 1001
by Inequality (3.9). We will go a little bit further to investigate this situation.
0. We may and do assume that P2 ≤ 1 and P3 ≤ 1.
1. If P7 ≥ 3 (resp. P6 ≥ 3, P5 ≥ 3), then K3 ≥ 5678 > 801 (resp. 601 ,501 )
by Corollary 3.1(with m0 = 4, and m1 = 7, 6, 5 respectively). So we
may assume P5, P6, P7 ≤ 2. Since P6 ≥ P4 + P2, we see that P2 = 0
and P6 = P4 = 2.
2. If P3 = 0, then n01,3 = P5−2 ≥ 0 implies P5 = 2. Now n51,4 = 3−σ5 ≥
0 gives σ5 ≤ 3. However n51,3 ≥ 0 implies σ5 ≥ 4, a contradiction. We
thus assume that P3 = 1 from now on.
3. We thus can make the following complete table for B(5) depending
on P5, σ5: No. P5 σ5 B(5) K3 ǫ + P7 1 1 0 {2 × (1, 2), (2, 5), 5 × (1, 4)} 1/20 4 2 1 1 {3 × (1, 2), (1, 3), 4 × (1, 4), (1, r)} 1/r − 1/6 4 3 2 1 {(1, 2), 2 × (2, 5), 3 × (1, 4), (1, r)} 1/r − 3/20 5 4 2 2 {2 × (1, 2), (2, 5), (1, 3), 2 × (1, 4), (1, r1), (1, r2)} 1/r1+ 1/r2− 11/30 5 5 2 3 {3 × (1, 2), 2 × (1, 3), (1, 4), (1, r1), (1, r2), (1, r3)} 1/r1+ r2+ r3− 7/12 5
4. By definition, one has σ5 ≤ ǫ ≤ 2σ5. Note that No. 1 is impossible
because ǫ = 0 but P7 ≤ 2 implies that ǫ ≥ 2, a contradiction. In No.
3, P5 = 2 implies P7 = 2 and hence ǫ = 3 > 2σ5, a contradiction.
In No. 2, one must have P7 = 2 and ǫ = 2 = 2σ5. Hence r ≥ 6.
Then it follows that K3 ≤ K3(B(5)) ≤ 0, a contradiction. Similarly,
in No. 4, K3(B(5)) > 0 only when r
1 = r2 = 5. But then ǫ = 2, a
contradiction.
5. It remains to consider No. 5. Note that K3(B(5)) > 0 only when
r1 = r2 = r3 = 5 and K3(B(5)) = 601. There are only finitely many
possible packings. Among them, we search for baskets with K3 ≥ 1
100.
It turns out there is only one new baskets
with K3(B
90) = 901.
Proposition 4.5. If P5 ≥ 2, then KX3 ≥ 1351 .
Proof. Similarly, we have K3
X ≥ 1351 unless P5 = 2, b = 0 and F a
(1, 0) surface, for which we have K3
X ≥ 1801 . Furthermore, we may
assume that Pm ≤ 2 for m = 6, 7, 8 by Corollary 3.1. It suffices to
consider: χ(OX) = 1, P2 = 0, P3 = 0, 1, P4 = 0, 1, P5 = P7 = 2 and
P4 ≤ P6 ≤ P8 ≤ 2.
We look at B(5) with K3 > 0 according to (P
3, P4, P6) and σ5. It
turns out that there is only one,
B(5) = {2 × (2, 5), 3 × (1, 3), (1, 4), (1, 6)} with K3(B(5)) = 1
60, given by (P3, P4, P6) = (1, 1, 2) and σ5 = 2. Now
P8 = 2 and hence B(7) = {2 × (2, 5), 2 × (1, 3), (2, 7), (1, 6)}. However K3(B(7)) = 1 210 < 1 180, which is impossible. Proposition 4.6. If P6 ≥ 2, then KX3 ≥ 2241 .
Proof. Similarly, we have K3
X ≥ 2241 unless P6 = 2, b = 0 and F a (1, 0)
surface, for which we have K3
X ≥ 2941 . Again, we may assume that
Pm ≤ 2 for m = 7, 8, 9, 10. Therefore, it remains to consider such a
situation that χ(OX) = 1, P2 = 0, P4 ≤ 1, P3 ≤ P5 ≤ 1, P7 ≤ P9 ≤ 2
and P8 = P10 = 2. According to the value of (P3, P4, P5) and σ5, we
have the following table.
No. (P3, P4, P5) σ5 B(5) K3 ǫ + P7 1 (0,0,0) 0 {5 × (1, 2), 4 × (1, 3), (1, 4)} 1/12 2 2 (0,0,1) 0 {3 × (1, 2), 2 ∗ (2, 5), 3 ∗ (1, 3)} 1/10 3 3 (0,1,0) 0 {6 ∗ (1, 2), (1, 3), 3 ∗ (1, 4)} 1/12 3 4 (0,1,1) 0 {4 ∗ (1, 2), 2 ∗ (2, 5), 2 ∗ (1, 4)} 1/10 4 5 (0,1,1) 1 {5 ∗ (1, 2), 1 ∗ (2, 5), (1, 3), (1, 4), (1, r)} 1/r − 7/60 4 6 (0,1,1) 2 {6 ∗ (1, 2), 2 ∗ (1, 3), (1, r1), (1, r2)} 1/r1+ 1/r2− 1/3 4 7 (1,0,1) 0 {(2, 5), 6 ∗ (1, 3), (1, 4)} 1/20 2 8 (1,0,1) 1 {(1, 2), 7 ∗ (1, 3), (1, r)} 1/r − 1/6 2 9 (1,1,1) 0 {(1, 2), (2, 5), 3 ∗ (1, 3), 3 ∗ (1, 4)} 1/20 3 10 (1,1,1) 1 {2 ∗ (1, 2), 4 ∗ (1, 3), 2 ∗ (1, 4), (1, r)} 1/r − 1/6 3
1. It is clear that No. 2, 3, 4, 9 are not allowed for ǫ = 0 and hence P7 ≥ 3.
2. In No. 1, 7, the baskets allow at most one packing at level 7, i,e, ǫ7 ≤ 1. However, P7 = 2 and P8 = 2 yield ǫ7 ≥ 2, a contradiction.
3. Consider No. 10. Since K3 = 1
r −
1
6 > 0, it follows that r = 5. So
ǫ = 1 and P7 = 2. Then ǫ7 = 2 and
B(7) = {2 × (1, 2), 2 × (1, 3), 2 × (2, 7), (1, 5)}.
This already implies ǫ8 = 0 and so we get P9 = 3, a contradiction.
4. Consider No. 8. Since K3 > 0, thus we get
Since B(5) allows no further packing, hence K3
X = 301 in this case.
5. Consider No. 5. Since K3 > 0, r = 6, 7, 8. It is easy to see that the
basket with the smallest volume and dominated by B(5) is
B210= {(7, 15), (2, 7), (1, 6)}
with K3 = 1
210. Thus K 3
X ≥ 2101 .
6. Finally Consider No. 6. Since K3 > 0, (r
1, r2) = (5, 5), (5, 6), (5, 7).
It is easy to see that the basket with the smallest volume and dominated by B(5) is B105 = {6 × (1, 2), 2 × (1, 3), (1, 5), (1, 7)} with K3 = 1 105. Thus K 3 X ≥ 1051 .
Note that, when δ(X) ≥ 7, we can utilize our explicit classification in [7, Section 3]. We shall omit some details to avoid unnecessary redundancy.
Proposition 4.7. If P7 ≥ 2, then KX3 ≥ 3361 .
Proof. Similarly, we have K3
X ≥ 3361 unless P7 = 2, b = 0, F a (1, 0)
surface and χ(OX) = 1. Again, we may assume that Pm ≤ 2 for
m = 8, 9. Hence P9 = 2 and P2 = 0.
By ǫ6 = 0, we have P4+P5+P6 = P3+2+ǫ. Hence (P3, P4, P5, P6) =
(0, 0, 1, 1), (0, 1, 0, 1), (0, 1, 1, 1) or (1, 1, 1, 1) which corresponds to Cases IV, V, VI, and VIII in [7, Section 3] respectively. The classification implies that, if K3
X < 3361 , then BX Bmin, where Bmin is a minimal
positive basket and belongs to one of the following:
(b1) B6,4 = {(1, 2), (6, 13), (1, 3), 2 × (1, 5)} with K3(B6,4) = 3901 and
P9(B6,4) = 3;
(b2) B6,6 = {3×(1, 2), (3, 7), (2, 5), (1, 4), (1, 6)} with K3(B6,6) = 4201
and P9(B6,4) = 3;
(b3) B8,3 = {2 × (2, 5), (1, 3), (3, 11), (1, 4)} with K3(B8,3) = 6601 .
Clearly, Case b1 can not happen because P9(BX) ≥ P9(Bmin) = 3.
In the Case b2, for the similar reason, BX 6= B6,6. Thus BX B60 :=
{4 × (1, 2), 2 × (2, 5), (1, 4), (1, 6)} and so K3
X ≥ K3(B60) = 601.
Finally, in Case b3, the proof of [7, Theorem 3.11] implies that BX 6=
B8,3 and BX B210 = {2 × (2, 5), (1, 3), (2, 7), 2 × (1, 4)} with KX3 ≥
K3(B
210) = 2101 . We have proved the statement.
It is now immediately to see the following consequences:
Corollary 4.8. (=Corollary 1.5) Let X be a minimal projective 3-fold of general type with K3
X < 3361 . Then δ(X) ≥ 8.
Proposition 4.9. Let X be a minimal projective 3-fold of general type. (1) If P8 ≥ 2, then KX3 ≥ 5041 .
(2) If P9 ≥ 2, then KX3 ≥ 6751 .
(4) If P11≥ 2, then KX3 ≥ 11881 .
(5) If P12≥ 2, then KX3 ≥ 15601
Proof. We only prove (1). Other statements can be proved similarly. When P8 ≥ 2, Table A1, Table A2, Inequalities (3.4). (3.6), Table
A3 and Table A4 imply K3
X ≥ 5041 unless we are in Subcase 3.4.4, for
which one has K3
X ≥ 4201 by [7, Theorem 1.2(2)] since χ(OX) = 1.
Propositions 4.2, 4.3, 4.4, 4.5, 4.6, 4.7 and 4.9 imply Theorem 4.1. An interesting by-product is the following:
Corollary 4.10. (=Corollary 1.7(1)) Let X be a minimal projective 3-fold of general type with pg(X) = 1. Then KX3 ≥ 751.
Proof. We distinguish the following cases. Case 1. P4 ≥ 3.
By Corollary 3.3, K3
X ≥ 1603 .
Case 2. P4 = 2.
We have K3
X ≥ 701 by Inequalities (3.4), (3.6) and Table A3 unless b = 0
and F is either a (1, 1) or a (1, 0) surface, for which we necessarily have h2(O
X) = 0 and thus χ(OX) = 0. Reid’s Riemann-Roch formula
implies P5 > P4 = 2. Now Corollary 3.1(with m0 = 4, m1 = 5) yields
K3 X ≥ 501 .
Case 3. P4 = 1.
Since pg(X) = 1, one has Pm > 0 for all m > 1. By [6, (3.10)], we have
P4+ P5+ P6 = 3P2+ P3+ P7+ ǫ ≥ 3P2+ P3 + P7.
If P4 = 1 (which implies P3 = P2 = 1), then we have
P5 ≥ (P7− P6) + 3 ≥ 3.
Then, from [6, (3.6)], n0
1,4 ≥ 0 implies χ(OX) ≥ 3. Due to our
previous result [5, Corollary 1.2] for irregular 3-folds, we may assume
q(X) = 0. Thus we have h2(OX) = χ(OX) ≥ 3. Take a sub-pencil
Λ of |5KX|. Then Λ induces a fibration f : X′ −→ Γ after Stein
factorization. Let F be the general fiber and F0 be the minimal model
of F .
Claim. K2
F0 ≥ 2.
Proof. Clearly we may write
f∗ωX′ = OΓ⊕ OΓ(e2) ⊕ · · · ⊕ OΓ(epg(F )−1)
with −2 ≤ ej ≤ −1 for all j, since pg(X′) = 1. Note that we have
h2(OX) = h1(f∗ωX′) + h0(R1f∗ωX′)
≤ (pg(F ) − 1) + h0(R1f∗ωX′).
If q(F ) > 0, we have K2
F0 ≥ 2 by the surface theory. If q(F ) = 0, we
have R1f
∗ωX′ = 0 and thus pg(F ) ≥ h2(OX) + 1 ≥ 4. Hence we have
If d5 ≥ 2, then we may set m1 = 5 and apply Inequality (3.10), which
gives K3 X ≥ 751 .
If d5 = 1, then |5KX′| and Λ are composed with the same pencil.
Thus we have θ5 ≥ 2 and Inequality (3.6) gives KX3 ≥ 24516.
5. Threefolds with δ(V ) ≥ 13
Let X be a minimal projective 3-fold of general type with δ(X) ≥ 13. Now we are in the natural position to classify baskets B(X) with
δ(X) ≥ 13. In fact, we have B12 B(X) B
min for certain minimal
positive basket Bminlisted in [7, Table C], where B12is also listed there.
However, as pointed out in [7, Proposition 4.5], our earlier classification in [7, Table C] is not clean since some minimal baskets in Table C are actually known to be “non-geometric”.
Recall that, by definition, a geometric weighted basket is a basket of a projective threefold of general type. Hence the following properties hold:
A. PmPn≤ Pm+n if Pm = 1 and n > 0.
B. Pm ≥ 0 for all m > 0.
C. K3 ≥ f (m
0) for some explicit function f (x) given in Sections 3
and 4 provided that Pm0 ≥ 2.
Indeed, if B12 violates one of A, B, C, then so does B(X). Therefore
B(X) is non-geometric. If Bmin is non-geometric (e.g. cases No. 3a, 5b,
10a, · · · , etc.), then we need to check all baskets between B12 and B min.
The following Table H consists of non-geometric baskets with δ ≥ 13. We keep the same notation as in Table C.
Table H N o. (P12, · · · , P24) (n1,2, n4,9, · · · , n1,5) or Bmin K3 Offending 3a (1, 0, 0, 1, 0, 0, 2, 0, 3, 1, 1, 1, 3) {(2, 5), (3, 8), ∗} ≻ {(5, 13), ∗} 3003017 P8P8> P16 5b (1, 0, 1, 2, 0, 0, 3, 0, 2, 1, 2, 2, 3) {(5, 13), (4, 15), ∗} 11701 P8P8> P16 8 (1, 0, 2, 1, 0, 1, 3, 1, 4, 3, 2, 2, 5) (7, 1, 0, 1, 0, 2, 0, 0, 6, 0, 2, 0, 0, 0, 1) 1 770 P6P10> P16 9 (1, 0, 2, −1, 1, 0, 2, 0, 1, 2, 1, 0, 2) (9, 0, 0, 2, 0, 0, 1, 1, 4, 0, 1, 0, 0, 1, 0) 55441 P15= −1
10a (1, 0, 2, 1, 2, −1, 2, 0, 2, 2, 1, 2, 4) {(4, 9), (3, 7), ∗} ≻ {(7, 16), ∗} 1 1680 P17= −1 11a (1, 0, 2, 0, 2, 0, 2, 2, 2, 1, 1, 1, 3) {(3, 8), (4, 11), ∗} ≻ {(7, 19), ∗} 26601 P8P14> P22 13 (1, 0, 3, −1, 1, 1, 3, 1, 3, 3, 3, 1, 4) (12, 0, 0, 2, 0, 2, 0, 2, 4, 0, 2, 0, 0, 1, 0) 34654 P15= −1 15a (1, 0, 3, 0, 1, 0, 2, 0, 3, 1, 1, 1, 4) {(4, 11), (1, 3), ∗} ≻ {(5, 14), ∗} 25201 P8P14> P22 15b (1, 0, 2, 0, 1, 0, 3, 0, 3, 2, 1, 1, 4) {(2, 5), (3, 8), ∗} ≻ {(5, 13), ∗} 3603623 P8P14> P22 15c (1, 0, 3, 1, 2, 0, 3, 1, 3, 2, 2, 2, 5) {(7, 16), (7, 19), ∗} 31 31920 P8P14> P22 16c (1, 0, 2, 1, 1, −1, 3, −1, 2, 2, 1, 1, 3) {{(5, 13), (7, 16)∗} 160163 P17= −1 18a (1, 0, 3, 0, 1, 0, 2, 1, 2, 2, 2, 1, 3) {(4, 11), (1, 3), ∗} ≻ {(5, 14), ∗} 30801 P6P11> P17 19 (1, 0, 2, 0, 1, 1, 3, 0, 2, 2, 2, 1, 3) (8, 0, 1, 1, 0, 1, 0, 1, 5, 0, 1, 0, 0, 1, 0) 2 3465 P9P14> P23 20a (1, 0, 1, 1, 1, 0, 3, −1, 2, 1, 0, 1, 3) {(2, 5), (3, 8), ∗} ≻ {(5, 13), ∗} 163801 P19= −1 21a (1, 1, 1, 1, 2, 0, 2, 1, 2, 1, 2, 2, 3) {(1, 3), (3, 10), ∗} ≻ {(4, 13), ∗} 46801 P8P9> P17 22 (1, 0, 1, 1, 1, 0, 2, 1, 3, 1, 1, 1, 3) (7, 1, 0, 1, 0, 1, 1, 0, 5, 1, 0, 0, 1, 0, 1) 1 9240 P8P9> P17 23a (1, 0, 2, 1, 2, 0, 2, 1, 3, 1, 2, 2, 3) {(4, 9), (3, 7), ∗} ≻ {(7, 16), ∗} 26401 P8P9> P17 24 (1, 0, 2, 0, 0, 1, 3, 0, 3, 2, 2, 0, 3) (10, 1, 0, 1, 0, 3, 0, 1, 6, 0, 2, 0, 0, 1, 0) 34651 P8P8> P16 26a (1, 0, 3, 1, 1, 1, 3, 0, 4, 1, 2, 2, 5) {(4, 11), (1, 3), ∗} ≻ {(5, 14), ∗} 1 1260 P9P10> P19 27.1 (1, 0, 2, 2, 1, 1, 5, 0, 4, 3, 3, 3, 6) {(2, 5), (3, 8), ∗} ≻ {(5, 13), ∗} 4504571 P9P10> P19 27.2 (1, 0, 2, 2, 1, 1, 5, −1, 3, 2, 2, 2, 4) {(2, 5), (5, 13), ∗} ≻ {(7, 18), ∗} 1 1386 P19= −1 27a (1, 0, 2, 2, 1, 1, 5, −1, 3, 2, 2, 2, 3) {(2, 5), (7, 18), ∗} ≻ {(9, 23), ∗} 1 1386 P19= −1 27b (1, 0, 2, 2, 1, 1, 5, −1, 3, 2, 2, 2, 5) {(5, 13), (5, 18), ∗} 11701 P19= −1 29a (1, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3) {(5, 14), (1, 3), ∗} ≻ {(6, 17), ∗} 1 5335 P9P14> P23 32b (1, 0, 3, 1, 1, 1, 3, 1, 3, 2, 3, 2, 4) {(4, 11), (1, 3), ∗} ≻ {(5, 14), ∗} 13861 P9P14> P23 33a (1, 1, 2, 0, 2, 1, 1, 1, 2, 2, 1, 2, 3) {(3, 10), (2, 7), ∗} ≻ {(5, 17), ∗} 28561 P6P16> P22 34b (1, 1, 2, 0, 1, 1, 3, 0, 3, 3, 1, 2, 4) {(2, 5), (3, 8), ∗} ≻ {(5, 13), ∗} 1 1170 P6P13> P19 39a (1, 1, 2, 1, 3, 0, 2, 1, 3, 2, 2, 3, 4) {(4, 9), (3, 7), ∗} ≻ {(7, 16), ∗} 16801 P6P16> P22 39b (1, 1, 2, 1, 3, 1, 2, 1, 3, 2, 2, 3, 5) {(3, 10), (2, 7), ∗} ≻ {(5, 17), ∗} 53554 P6P16> P22 40.1 (1, 1, 2, 1, 2, 1, 4, 0, 4, 3, 2, 3, 6) {(2, 5), (3, 8), ∗} ≻ {(5, 13), ∗} 41 32760 P6P13> P19 40a (1, 1, 2, 1, 2, 1, 4, −1, 3, 2, 1, 2, 4) {(4, 10), (3, 8), ∗} ≻ {(7, 18), ∗} 25201 P6P13> P19 40b (1, 1, 2, 1, 2, 1, 4, 0, 4, 3, 1, 2, 5) {(2, 5), (6, 16), ∗} ≻ {(8, 21), ∗} 12601 P6P13> P19 43a (1, 1, 3, 0, 2, 1, 2, 1, 3, 2, 2, 2, 4) {(4, 11), (1, 3), ∗} ≻ {(5, 14), ∗} 1 2520 P7P8> P15 43b (1, 1, 2, 0, 2, 1, 3, 1, 3, 3, 2, 2, 4) {(2, 5), (3, 8), ∗} ≻ {(5, 13), ∗} 3603623 P7P8> P15 44a (1, 1, 2, 1, 2, 1, 4, 1, 3, 4, 2, 2, 4) {(2, 5), (6, 16), ∗} ≻ {(8, 21), ∗} 13861 P7P18> P25= 3 44b (1, 1, 2, 1, 2, 0, 3, 0, 2, 3, 2, 2, 3) {(7, 16), (5, 13), ∗} 3 16016 P7P10> P17 46a (1, 1, 1, 1, 2, 1, 3, 0, 3, 1, 1, 2, 3) {(2, 5), (3, 8), ∗} ≻ {(5, 13), ∗} 163801 P9P10> P19 50a (1, 1, 3, 1, 2, 2, 3, 1, 4, 2, 3, 3, 5) {(4, 11), (1, 3), ∗} ≻ {(5, 14), ∗} 1 1260 P7P14> P21 51a (1, 1, 2, 2, 2, 2, 5, 0, 3, 3, 3, 3, 4) {(4, 10), (3, 8), ∗} ≻ {(7, 18), ∗} 1 1386 P6P13> P19 51b (1, 1, 2, 2, 2, 2, 5, 0, 3, 3, 3, 3, 5) {(5, 13), (5, 18), ∗} 11701 P6P13> P19 52a (1, 1, 2, 1, 1, 0, 2, 1, 2, 2, 1, 2, 3) {(2, 5), (3, 8), ∗} ≻ {(5, 13), ∗} 1 2184 P5P12> P17 56a (1, 1, 2, 2, 1, 1, 2, 1, 3, 2, 2, 3, 3) {(4, 9), (3, 7), ∗} ≻ {(7, 16), ∗} 16801 P5P14> P19 57 (1, 0, 2, 2, 0, 1, 3, 1, 3, 2, 2, 2, 3) (3, 0, 1, 2, 0, 5, 0, 0, 4, 0, 0, 1, 0, 0, 0) 13861 P7P9> P16 58a (1, 1, 2, 2, 2, 0, 2, 1, 3, 2, 2, 3, 4) {(4, 9), (3, 7), ∗} ≻ {(7, 16), ∗} 1 1680 P5P12> P17 59a (1, 1, 2, 1, 2, 1, 2, 3, 2, 2, 2, 2, 3) {(3, 8), (4, 11), ∗} ≻ {(7, 19), ∗} 26601 Item C 60a (1, 1, 1, 2, 1, 1, 3, 0, 3, 1, 1, 2, 3) {(2, 5), (3, 8), ∗} ≻ {(5, 13), ∗} 163801 P9P10> P19 61 (1, 1, 1, 2, 1, 1, 2, 2, 3, 2, 2, 2, 3) (0, 1, 0, 1, 0, 3, 1, 0, 2, 0, 0, 0, 1, 0, 0) 1 9240 Item C 62a (1, 1, 2, 2, 2, 1, 2, 2, 3, 2, 3, 3, 3) {(4, 9), (3, 7), ∗} ≻ {(7, 16), ∗} 26401 Item C 63 (1, 1, 3, 1, 2, 1, 3, 2, 3, 3, 2, 2, 4) (5, 0, 1, 2, 0, 1, 1, 1, 3, 0, 1, 0, 0, 0, 1) 55441 Item C
By eliminating non-geometric baskets, we obtain a shorter list of baskets, listed in Table F-0, F-1, F-2 in the Appendix. We summarize some observations from the Tables.
Theorem 5.1. (=Theorem 1.4) Let X be a minimal projective 3-fold of general type with the weighted basket B(X) := {BX, P2, χ(OX)}. If
δ(X) ≥ 13, then P2 = 0 and B(X) belongs to one of the types listed in
Tables F–0∼ F–2 in Appendix. Furthermore, the following holds: (1) δ(X) = 18 if and only if B(X) = {B2a, 0, 2} (see Table F–0 for
B2a) with KX3 = 11701 .
(3) δ(X) = 15 if and only if B(X) is among one of the cases in
Table F–1. One has K3
X ≥ 13861 .
(4) δ(X) = 14 if and only if B(X) is among one of the cases in
Table F–2. One has K3
X ≥ 16801 .
(5) δ(X) = 13 if and only if B(X) = {B41, 0, 2} (see Table F–0 for
B41) with KX3 = 2521 .
Theorem 4.1, Theorem 5.1 and [11, Theorem 1.4] imply the following: Corollary 5.2. (=Theorem 1.6(2)) Let X be a minimal projective 3-fold of general type. Then K3
X ≥ 16801 , and equality holds if and only if
χ(OX) = 2, P2 = 0 and BX = B7a or BX = B36a (cf. Table F–2).
Theorem 5.1, together with the explicit calculation, also implies the following:
Corollary 5.3. Let X be a minimal projective 3-fold of general type. Then,
(1) if δ(X) = 13, Pm > 0 for all m ≥ 10;
(2) if δ(X) = 14, 15, 18, Pm > 0 for all m ≥ 20.
6. Birationality
Theorem 6.1. Let X be a minimal projective 3-fold of general type. If δ(X) = 18, then Φm is birational for all m ≥ 61.
Proof. Set m0 = 18. By Theorem 5.1, we know that BX = B2a, P2 = 0,
χ(OX) = 2, P19= 0, P24= 3 and KX3 = 11701 . By [5, Corollary 1.2], we
see q(X) = 0. Thus |18KX| induces a fibration f : X′ −→ Γ ∼= P1. We
have h2(O
X′) = h2(OX) = 1. Pick a general fiber F . Since P19(X) =
P19(B2a) = 0, we have H0(X′, KX′ + F ) = 0.
Claim 6.1.1. pg(F ) = 1.
Proof. Since χ(OX′) > 1, we have pg(F ) > 0 by [7, Lemma 2.32]. On
the other hand, we have the long exact sequence:
H0(X′, KX′+F ) −→ H0(F, KF) −→ H1(X′, KX′) −→ H1(X′, KX′+F )
which implies h0(K
F) ≤ h1(X′, KX′) = h2(OX′) = 1. Thus we get
pg(F ) = 1.
We have Pm > 0 for all m ≥ 20 by Corollary 5.3 (2). Consider the
linear systems
|KX′+ ⌈nπ∗(KX)⌉ + F ||(n + 19)KX′|.
Clearly |(n + 19)KX′| distinguish different general fibers F as long as
n ≥ 19. By Kawamata-Viehweg vanishing,
|KX′ + ⌈nπ∗(KX)⌉ + F ||F = |KF + ⌈nπ∗(KX)⌉|F|
|KF + ⌈Ln⌉|
Claim 6.1.2. L2
n > 8 whenever n ≥ 42.
Proof. Since pg(F ) = 1, we are in Subcase 3.4.1 or Subcase 3.4.3.
Let us consider Subcase 3.4.1 (i.e. K2
F0 ≥ 2) first. We have (π∗(KX)|F)2 ≥ 1 192K 2 F0 ≥ 2 192
by Lemma 2.1(ii). Thus L2
n > 8 whenever n > 38.
If K2
F0 = 1, we shall estimate L
2
nin an alternative way. Suppose that
|24KX′| and |18KX′| are not composed with the same pencil. Take
|G| := |M24|F|. Pick a generic irreducible element C of |G|. Then we
have ξ = (π∗(K
X)|F · C) ≥ 192 by Lemma 2.4. Thus (π∗(KX)|F)2 ≥ 1
24ξ ≥ 1
12·19. Since r(X) = 2340 and r(X)(π∗(KX)|F)2 is an integer, we
see (π∗(K
X)|F)2 ≥ 234011 . So we have L2n> 8 whenever n ≥ 42.
Assume that |24KX′| and |18KX′| are composed with the same
pen-cil. Since P24 = 3, we may set m0 = 24 and Λ = |24KX′|. We have
θ = 2. The argument in Subcase 3.4.3 implies that (π∗(KX)|F)2 ≥ 4θ2 ( ˜m0+ θ)(3m0+ 4θ) = 1 130. We have L2 n > 8 whenever n ≥ 33.
For very general curves ˜C on F , one has (Ln· ˜C) ≥ n 19(σ ∗(K F0) · ˜C) ≥ 2n 19
by Lemma 2.5. Therefore, (Ln · ˜C) ≥ 4 for n ≥ 38. Lemma 2.3
implies that |KF + ⌈Ln⌉| gives a birational map for n ≥ 42. Thus Φm
is birational for all m ≥ 61.
Theorem 6.2. Let X be a minimal projective 3-fold of general type. If δ(X) ≤ 15, then Φm is birational for all m ≥ 56.
Proof. Set m0 = δ(X). By considering a sub-pencil Λ of |m0KX|, we
may always assume that we have an induced fibration f : X′ −→ Γ
onto a curve Γ. By Chen-Hacon [9], we may assume q(X) = 0. Thus Γ ∼= P1. By [7, Corollary 3.13] and [7, Lemma 2.32], we know that δ(X) ≤ 10 as long as F is a (1, 0) surface. Therefore it suffices to consider the following 3 cases:
1. δ(X) ≤ 15 and F is a (1, 2) surface.
2. δ(X) ≤ 15 and F is neither a (1, 2) surface nor a (1, 0) surface. 3. δ(X) ≤ 10 and F is a (1, 0) surface.
Case 1. Without losing of generality, let us assume δ(X) = 15. Take |G| to be the moving part of |KF|. Then, by Table A3, we have ξ ≥ 111 .
We have m0 = 15 and β 7→ 161. So αm > 2 whenever m ≥ 55. By
m ≥ 35. On the other hand, Kawamata-Viehweg vanishing and Lemma 2.1 imply the following, whenever m ≥ 49,
|mKX′||F |KX′ + ⌈(m − 16)π∗(KX)⌉ + F ||F
|KF + ⌈(m − 16)π∗(KX)|F⌉
|(KF + ⌈Qm⌉ + C) + C|
where Qm is a nef and big Q-divisor. Thus, by [7, Lemma 2.17], Φm
distinguishes different generic curves C for m ≥ 49. Finally Theorem 2.7 implies that Φm is birational for all m ≥ 55.
Case 2. Still assume δ(X) = 15. Parallel to the respective argument in the proof of Theorem 6.1, one knows that |mKX′| distingishes different
general fibers F for m ≥ 35. By the surface theory, we see that F is either a surface with K2
F0 ≥ 2 or a (1, 1) surface. We want to study
the linear system |KF + ⌈Ln⌉|. In fact, by the estimation in Subcase
3.4.1 and Table A4, we have L2
n≥ n
2
32·6 > 8 whenever n ≥ 40. Similarly
we have (Ln· ˜C) ≥ 4 for all n ≥ 32 and for all curves ˜C on F passing
through very general points. By Lemma 2.3, we see that |KF + ⌈Ln⌉|
gives a birational map for all n ≥ 40. Similar to what discussed in the proof of Theorem 6.1, we have proved that Φm is birational for all
m ≥ n + 16 ≥ 56.
Case 3. When δ(X) ≤ 10, we have much better birationality result even though F is a (1, 0) surface. In fact, parallel argument shows that Φm is birational for all m ≥ 39. The proof is more or less similar to
above ones. We leave it as an exercise to interested readers.
Theorems 5.1, 6.1, and 6.2 imply Theorem 1.6 (2).
7. Threefolds with δ(V ) = 2
This section is devoted to classifying minimal projective threefolds of general type with δ(X) = 2, that is, pg(X) ≤ 1 and P2(X) ≥ 2.
Assume that P2 ≥ 2. We first recall the following known results:
(a) If d2 = 3, then Φm is birational for all m ≥ 7 by [7, Theorem
2.20].
(b) If d2 = 2, Φm is birational for all m ≥ 10 by [7, Theorem 2.22].
(c) If q(X) > 0, then Φm is birational for all m ≥ 7 by Chen–Hacon
[9] and for m = 6 by Chen-Chen-Jiang [8].
The purpose of this section is to prove that Φm is birational for m ≥
11 and classify threefolds such that Φ10 is not birational. Therefore,
we may and do assume that q(X) = 0, d2 = 1 and b = g(Γ) = 0. Let
F be the general fiber of the induced fibration f : X′ → P1 from Φ 2.
7.1. Birationality of Φm for m ≥ 11.
Lemma 7.1. |mKX′| distinguishes different general fibers of f for all
m ≥ 9.
Proof. When pg(F ) > 0, by [7, Proposition 2.15 (i)], one has Pk > 0
for k ≥ 7. Thus, for all m ≥ 9, mKX′ ≥ F , hence |mKX′| distinguishes
different general fibers of f .
When pg(F ) = 0, one has χ(OX) ≤ 1 (cf. [7, Lemma 2.32]). By
[7, Lemma 3.2], one has P5 ≥ P2 > 0. Then clearly Pk > 0 for all
k ≥ 5. Thus, for all m ≥ 7, mKX′ ≥ F and hence |mKX′| distinguishes
different general fibers of f .
Proposition 7.2. Assume P2(X) ≥ 2, q(X) = 0, d2 = 1 and F is not
a (1, 2) surface. Then Φm is birational for all m ≥ 10.
Proof. Set Ln := nπ∗(KX)|F which is a nef and big Q-divisor on F .
Kawamata-Viehweg vanishing gives the following surjective map: H0(X′, KX′ + ⌈nπ∗(KX)⌉ + F ) −→ H0(F, KF + ⌈nπ∗(KX)⌉|F).
Together with Lemma 7.1, it is sufficient to prove that |KF + ⌈Ln⌉|
gives a birational map for n ≥ 7 because
|(n + 3)KX′| |KX′ + ⌈nπ∗(KX)⌉ + F |.
Claim 7.2.1. If K2
F0 ≥ 2 or F0 is of type (1, 0), then |KF + ⌈Ln⌉| is
birational for n ≥ 7.
First of all, for any curve ˜C ⊂ F passing through very general points of F , we estimate (Ln· ˜C) for n ≥ 7. Clearly we have g( ˜C) ≥ 2. Set
m0 = 2 and Λ = |2KX′|. By Lemma 2.1 and Lemma 2.5, we have
(Ln· ˜C) ≥ 7(π∗(KX)|F · ˜C) ≥ 7 3(σ ∗(K F0) · ˜C) > 4. If K2 F0 ≥ 2, then we have L2n≥ 49(π∗(K X)|F)2 ≥ 49( 1 3σ ∗(K F0)) 2 ≥ 98 9 > 8.
If F0 is a (1, 0) surface, we have P4 ≥ 2P2 ≥ 4 since χ(OX) ≤ 1.
When d4 ≥ 2, we set m0 = 2, Λ = |2KX′| and |G| = |M4|F|. Then
β = 1 4, ξ ≥ 1 3(σ ∗(K F0) · C) ≥ 2 3 and so L 2 n ≥ 496 > 8.
When d4 = 1, we set m0 = 4 and Λ = |4KX′|. Clearly |2KX′| and
|4KX′| induce the same fibration f . Take |G| = |2σ∗(KF0)|. Since θ ≥
3, we have β ≥ 3 14 by Lemma 2.1. Thus ξ ≥ 6 7 and so L 2 n ≥ 49·143 · 6 7 > 8.
By Lemma 2.3, the Claim follows.
Claim 7.2.2. If F0 is a (1, 1) surface, then |KF + ⌈Ln⌉| is birational
for n ≥ 7.
Following the similar argument as above, it is easy to see that L2
n≥
64
7 > 8 and (Ln· ˜C) ≥ 4 for all n ≥ 8. We consider the linear system
base point free. Pick a generic irreducible element C ∈ |2σ∗(K F0))|.
Since OΓ(1) ֒→ f∗ωX′, we have f∗ωX2′/Γ ֒→ f∗ω10X′. The semi-positivity
implies that f∗ωX2′/Γ is generated by global sections, which directly
implies 10KX′|F ≥ C. Thus Φ10 distinguishes different C. By Lemma
2.1, we have 6π∗(K
X)|F ≡ C + H6 for an effective Q-divisor H6 on F .
Thus the vanishing theorem implies
|KF + ⌈7π∗(KX)|F − H6⌉||C = |KC+ D|
with deg(D) ≥ 2(⌈7π∗(K
X)|F − C − H6⌉ · σ∗(KF0)) ≥ 2. Since C
is non-hyperelliptic, |KC + D| gives a birational map. Thus |KF +
⌈7π∗(K
X)|F⌉| is birational.
Proposition 7.3. Assume P2(X) ≥ 2, q(X) = 0, d2 = 1 and F a
(1, 2) surface. Then Φm is birational for all m ≥ 11.
Proof. Take |G| to be the moving part of |σ∗(K
F0)|. Modulo birational
modifications, we may assume that |G| is base point free. Pick a generic irreducible element C of |G|. It is also known that g = 2.
Claim 7.3.1 The linear system |mKX′| distinguishes different general
members of |G| for m ≥ 9.
Proof. Clearly |G| is composed with a rational pencil since q(F ) = 0. We shall prove |mKX′||F |G| and thus the statement follows. In fact,
by Lemma 2.1, we have
3π∗(KX) ≡ σ∗(KF0) + H3
for an effective Q-divisor H3 on F . Thus, for m ≥ 10,
Qm := (m − 3)π∗(KX)|F − 2H3− 2σ∗(KF0) ≡ (m − 9)π
∗(K X)|F
is nef and big. It follows that KF + ⌈Qm⌉ + σ∗(KF0) > 0 by [7, Lemma
2.14]. We thus have the following:
|mKX′||F |KX′+ F + ⌈(m − 3)π∗(KX)⌉||F = |KF + ⌈(m − 3)π∗(KX)⌉|F| |KF + ⌈(m − 3)π∗(KX)|F − 2H3⌉| = |(KF + ⌈Qm⌉ + σ∗(KF0)) + σ ∗(K F0)| |σ∗(KF0)| |G|
where the first equality follows from the Kawamata-Viehweg vanishing ([19, 33]). Therefore, |mKX′| distinguishes general members of |G| for
m ≥ 10. Moreover, for m = 9,
|9KX′||F |5KX′||F |KX′+ ⌈2π∗(KX)⌉ + F ||F
= |KF + ⌈2π∗(KX)⌉|F| |G|
where the equality is again due to Kawamata-Viehweg vanishing. Hence |9KX′| distinguishes general members of |G| as well, which asserts the
From Table A3, one has ξ ≥ 1
2. Take m ≥ 11, then αm ≥ 5
2 > 2. This
means that |mKX′||C distinguishes points on C. Thus, by Theorem 2.7
and Claim 7.3.1, Φm is birational for all m ≥ 11.
Now Theorem 1.8.1 follows from Proposition 7.2 and Proposition 7.3. That is, if P2 ≥ 2, then Φm is birational for m ≥ 11.
If either ξ > 12 or β > 13 then α10 > 2. Hence the following
conse-quence is immediate.
Corollary 7.4. Let X be a minimal projective 3-fold of general type. Assume P2(X) ≥ 2, q(X) = 0, d2 = 1 and F0 a (1, 2) surface. If either
ξ > 12 or β > 13 or P2 > 2, then Φ10 is birational.
Proposition 7.2, Proposition 7.3 and Corollary 7.4 also imply the following:
Corollary 7.5. Let X be a minimal projective 3-fold of general type. Assume P2 ≥ 2 and Φ10 is not birational. Then P2 = 2, q(X) = 0 and
|2KX′| is composed with a rational pencil of (1, 2) surfaces.
7.2. Classification. In the rest of this section, we classify minimal 3-folds X of general type which satisfy the following assumptions:
(♯) P2(X) = 2 and Φ10 is not birational.
Note that Corollary 7.5 implies that |2KX| induces a fibration f :
X′ −→ P1 with the general fiber F a (1, 2) surface.
Lemma 7.6. If X satisfies (♯), then 0 ≤ χ(OX) ≤ 3.
Proof. Note that the general fiber F of f is a (1,2) surface. Since
q(F ) = 0, we have q(X) = 0, h2(O
X) = h1(P1, f∗ωX′) and pg(X) =
h0(f
∗ωX′). Since P2(X) = 2 implies pg(X) ≤ 1, we see χ(OX) ≥ 0. By
Fujita’s semi-positivity([16]), we have χ(OX) ≤ 3.
Theorem 7.7. Let X be a minimal projective 3-fold of general type. Assume P2 = 2, q(X) = 0 and F a (1, 2) surface. Then Φ10is birational
under one of the following conditions: (1) P3 ≥ 4;
(2) P4 ≥ 6;
(3) P5 ≥ 8;
(4) P6 ≥ 14.
Proof. We set m0 = 2. Pick a general fiber F of f : X′ −→ Γ and a
generic irreducible element C of |G| := Mov|σ∗(K
F0)| on F . For m1 =
3, 4, 5 and 6, we have Pm1 ≥ 4. Modulo further birational modifications
to π, we may assume that the moving part |Mm1| of |m1KX′| is base
point free. We consider the following natural maps: H0(X′, Sm1)
µm1
−→ H0(F, Sm1|F)
νm1
−→ H0(C, Sm1|C)
Let Mov|Sm1|F| be the moving part of |Sm1|F| and let Tm1 be a
general element in Mov|Sm1|F| when h
0(F, S
m1|F) > 1. Clearly
(Sm1 · C)X′ ≥ (Tm1 · C)F ≥ 0.
Since F and C are general, both µm1 and νm1 are non-zero maps. In
particular, h0(F, S
m1|F) > 0 and h
0(C, S
m1|C) > 0.
Let F(r) be a general element in Mov|Sm1− rF | if h
0(S
m1− rF ) ≥ 2.
Let C(r) be a general element in Mov|Tm1 − rC| if h
0(T
m1 − rC) ≥ 2.
Replace X′ by its birational modification, we may and do assume that
Mov|Sm1 − rF | is free. Clearly, for 0 < r ≤ h0(X′,Sm1) h0(F,S m1|F), we have h0(X′, S m1 − rF ) ≥ h 0(X′, S m1) − r · h 0(F, S m1|F). (7.1) Claim 7.7.1. If (Tm1 · C) ≤ 1, then (Tm1· C) = 0.
Proof. In fact, if |Tm1| 6= ∅ and |Tm1| is not composed of the same pencil
as that of |C|, then Φ|Tm1|(C) is a curve and so h0(C, Tm1|C) ≥ 2. Note
that g(C) = 2. The Riemann-Roch theorem and the Clifford theorem imply that (Tm1 · C) = deg(Tm1|C) ≥ 2, a contradiction. Hence either
|Tm1| is composed of the same pencil as that of |C| on F or |Tm1| = ∅.
Claim 7.7.1 now follows.
Claim 7.7.2. Keep the same notation as above. Then Φ10 is birational
under one of the following conditions: (1) (Tm1 · C) > m1 2 ; (2) Tm1 · C = 0 and h 0(F, T m1) > 1 + m1 3 ;
(3) Tm1 ≥ tC for some rational number t >
m1 3 ; (4) either |Tm1| = ∅ and Pm1 > 1+ m1 2 or |Tm1| 6= ∅ and ⌊ Pm1−1 h0(F,T m1)⌋ > m1 2 .
(5) F(r) (resp. C(r)) is algebraically equivalent to F (resp. C) and r+1 m1 > 1 2 (resp. r+1 m1 > 1 3). Proof. If (Tm1 · C) > m1 2 , then ξ ≥ 1 m1(Sm1 · C) ≥ 1 m1(Tm1 · C) > 1 2.
Then Corollary 7.4 implies that Φ10 is birational, which proves (1).
Now we prove (4). We claim that we have m1π∗(KX) ≥ Sm1 ≥ rF
for an integer r > m1
2 . In fact, when |Tm1| = ∅, |Sm1| is composed of
the same pencil as that of |F | and we may take r := Pm1 − 1. When
|Tm1| 6= ∅, we may take r = ⌊
Pm1−1 h0(F,T
m1)⌋ and then Sm1 ≥ rF since
dim im(µm1) ≤ h
0(F, T
m1). Then Lemma 2.1 implies β ≥
r m1+r >
1 3.
So Φ10 is birational by Corollary 7.4, which asserts (4).
Since m1π∗(KX)|F ≥ Tm1 ≥ tC, we have β >
1
3 and Φ10 is birational
